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%TCIDATA{Created=Mon Jun 07 15:27:51 2004}
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\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\begin{document}
\ La funci\'{o}n $\ f(x)=\sin(2x)$ es estrictamente positiva en\newline\qquad
a) $\left\{  \left(  \dfrac{-(k+4)\pi}{4},\dfrac{-(k+2)\pi}{4}\right)  \text{
},\left(  \dfrac{k\pi}{4},\dfrac{(k+2)\pi}{4}\right)  \text{con }%
k=0,4,8,12,...\right\}  $\newline\qquad b) $\left\{  \left[  \dfrac{-k\pi}%
{4},\dfrac{-(k-1)\pi}{4}\right]  \text{ },\left[  \dfrac{k\pi}{4}%
,\dfrac{(k+1)\pi}{4}\right]  \text{con }k=0,2,4,6,...\right\}  $\newline\qquad
c) $\left\{  \left(  \dfrac{-k\pi}{4},\dfrac{-(k-1)\pi}{4}\right)  \text{
},\left(  \dfrac{k\pi}{4},\dfrac{(k+1)\pi}{4}\right)  \text{con }%
k=0,1,2,3,...\right\}  $\newline\qquad d) $\left\{  \left(  \dfrac{-k\pi}%
{2},\dfrac{-(k-1)\pi}{2}\right)  \text{ },\left(  \dfrac{k\pi}{2}%
,\dfrac{(k+1)\pi}{2}\right)  \text{con }k=0,2,4,6,...\right\}  $

\ La funci\'{o}n $\ f(x)=\cos(2x)$ es estrictamente positiva en\newline\qquad
a) $\left\{  \left(  \dfrac{-k\pi}{4},\dfrac{-(k-2)\pi}{4}\right)  \text{
},\left(  \dfrac{(k-2)\pi}{4},\dfrac{k\pi}{4}\right)  \text{con }%
k=1,5,9,13,17,...\right\}  $\newline\qquad b) $\left\{  \left[  \dfrac{-k\pi
}{4},\dfrac{-(k-2)\pi}{4}\right]  \text{ },\left[  \dfrac{(k-2)\pi}{4}%
,\dfrac{k\pi}{4}\right]  \text{con }k=1,5,9,13,17,...\right\}  $\newline\qquad
c) $\left\{  \left(  \dfrac{-k\pi}{4},\dfrac{-(k-2)\pi}{4}\right)  \text{
},\left(  \dfrac{(k-2)\pi}{4},\dfrac{k\pi}{4}\right)  \text{con }%
k=0,1,2,3,4,5...\right\}  $\newline\qquad d) $\left\{  \left(  \dfrac{-k\pi
}{2},\dfrac{-(k-1)\pi}{2}\right)  \text{ },\left(  \dfrac{k\pi}{2}%
,\dfrac{(k+1)\pi}{2}\right)  \text{con }k=0,2,4,6,8,10...\right\}  $

\ La funci\'{o}n $\ f(x)=\cos(2x)$ es estrictamente negativa en\newline a)
$\left\{  \left(  \dfrac{-k\pi}{4},\dfrac{-(k-2)\pi}{4}\right)  \text{
},\left(  \dfrac{(k-2)\pi}{4},\dfrac{k\pi}{4}\right)  \text{con }%
k=3,7,11,15,19,...\right\}  $\newline b) $\left\{  \left[  \dfrac{-k\pi}%
{4},\dfrac{-(k-2)\pi}{4}\right]  \text{ },\left[  \dfrac{(k-2)\pi}{4}%
,\dfrac{k\pi}{4}\right]  \text{con }k=3,7,11,15,19,...\right\}  $\newline c)
$\left\{  \left(  \dfrac{-k\pi}{4},\dfrac{-(k-2)\pi}{4}\right)  \text{
},\left(  \dfrac{(k-2)\pi}{4},\dfrac{k\pi}{4}\right)  \text{con }%
k=2,4,6,8,10,12...\right\}  $\newline d) $\left\{  \left(  \dfrac{-k\pi}%
{2},\dfrac{-(k-1)\pi}{2}\right)  \text{ },\left(  \dfrac{k\pi}{2}%
,\dfrac{(k+1)\pi}{2}\right)  \text{con }k=0,2,4,6,8,10...\right\}  $

\ La funci\'{o}n $\ f(x)=\sin(2x)$ es estrictamente negativa en\newline\qquad
a) $\left\{  \left(  \dfrac{-k\pi}{4},\dfrac{-(k-2)\pi}{4}\right)  \text{
},\left(  \dfrac{k\pi}{4},\dfrac{(k+2)\pi}{4}\right)  \text{con }%
k=2,6,10,14,...\right\}  $\newline\qquad b) $\left\{  \left[  \dfrac{-k\pi}%
{4},\dfrac{-(k-1)\pi}{4}\right]  \text{ },\left[  \dfrac{k\pi}{4}%
,\dfrac{(k+1)\pi}{4}\right]  \text{con }k=2,6,10,14,...\right\}  $%
\newline\qquad c) $\left\{  \left(  \dfrac{-k\pi}{4},\dfrac{-(k-1)\pi}%
{4}\right)  \text{ },\left(  \dfrac{k\pi}{4},\dfrac{(k+1)\pi}{4}\right)
\text{con }k=2,4,6,8,10...\right\}  $\newline\qquad d) $\left\{  \left(
\dfrac{-k\pi}{2},\dfrac{-(k-1)\pi}{2}\right)  \text{ },\left(  \dfrac{k\pi}%
{2},\dfrac{(k+1)\pi}{2}\right)  \text{con }k=0,2,4,6,...\right\}  $

\ La funci\'{o}n $\ f(x)=\cos(x)$ es estrictamente positiva en\newline\qquad
a) $\left\{  \left(  \dfrac{-k\pi}{2},\dfrac{-(k-2)\pi}{2}\right)  \text{
},\left(  \dfrac{(k-2)\pi}{2},\dfrac{k\pi}{2}\right)  \text{con }%
k=1,5,9,13,17,...\right\}  $\newline\qquad b) $\left\{  \left[  \dfrac{-k\pi
}{2},\dfrac{-(k-2)\pi}{4}\right]  \text{ },\left[  \dfrac{(k-2)\pi}{2}%
,\dfrac{k\pi}{2}\right]  \text{con }k=1,5,9,13,17,...\right\}  $\newline\qquad
c) $\left\{  \left(  \dfrac{-k\pi}{4},\dfrac{-(k-2)\pi}{4}\right)  \text{
},\left(  \dfrac{(k-2)\pi}{4},\dfrac{k\pi}{4}\right)  \text{con }%
k=0,1,2,3,4,5...\right\}  $\newline\qquad d) $\left\{  \left(  \dfrac{-k\pi
}{2},\dfrac{-(k-1)\pi}{2}\right)  \text{ },\left(  \dfrac{k\pi}{2}%
,\dfrac{(k+1)\pi}{2}\right)  \text{con }k=0,2,4,6,8,10...\right\}  $

\ La funci\'{o}n $\ f(x)=\cos\left(  \dfrac{1}{2}x\right)  $ es estrictamente
positiva en \newline\qquad a) $\left\{  (-\pi,\pi),\left(  -(k+\dfrac{5}%
{2})2\pi,-(k+\dfrac{3}{2})2\pi\right)  ,\left(  (k+\dfrac{3}{2})2\pi
,(k+\dfrac{5}{2})2\pi\right)  \text{ con }k=0,2,4,6,...\right\}  $%
\newline\qquad b) $\left\{  [-\pi,\pi],\left[  -(k+\dfrac{5}{2})2\pi
,-(k+\dfrac{3}{2})2\pi\right]  ,\left[  (k+\dfrac{3}{2})2\pi,(k+\dfrac{5}%
{2})2\pi\right]  \text{ con }k=0,1,2,3,...\right\}  $\newline\qquad c)
$\left\{  (-\pi,\pi),\left(  -(k+\dfrac{5}{4})2\pi,-(k+\dfrac{3}{4}%
)2\pi\right)  ,\left(  (k+\dfrac{3}{4})2\pi,(k+\dfrac{5}{4})2\pi\right)
\text{ con }k=0,4,8,12,...\right\}  $\newline\qquad d) $\left\{  (-\pi
,\pi),\left(  -(k+\dfrac{5}{4})2\pi,-(k+\dfrac{3}{4})2\pi\right)  ,\left(
(k+\dfrac{3}{4})2\pi,(k+\dfrac{5}{4})2\pi\right)  \text{ con }%
k=0,1,2,3,...\right\}  $

\ La funci\'{o}n $\ f(x)=\cos(3x)$ es estrictamente positiva en \newline%
\qquad\medskip a) $\left\{  \left(  -\dfrac{\pi}{6},\dfrac{\pi}{6}\right)
,\left(  \dfrac{-(k+15)\pi}{8},\dfrac{-(k+9)\pi}{8}\right)  \text{ },\left(
\dfrac{(k+9)\pi}{8},\dfrac{(k+15)\pi}{8}\right)  \text{con }%
k=0,12,24,36,...\right\}  $\newline\qquad b) $\left\{  \left(  -\dfrac{\pi}%
{6},\dfrac{\pi}{6}\right)  ,\left(  \dfrac{-(k+15)\pi}{2},\dfrac{-(k+9)\pi}%
{2}\right)  \text{ },\left(  \dfrac{(k+9)\pi}{2},\dfrac{(k+15)\pi}{2}\right)
\text{con }k=0,1,2,3,...\right\}  $\newline\qquad c)$\left\{  \left(
-\dfrac{\pi}{6},\dfrac{\pi}{6}\right)  ,\left(  \dfrac{-(k+15)\pi}{2}%
,\dfrac{-(k+9)\pi}{2}\right)  \text{ },\left(  \dfrac{(k+9)\pi}{2}%
,\dfrac{(k+15)\pi}{2}\right)  \text{con }k=0,2,4,6,...\right\}  $
\newline\qquad d) $\left\{  \left[  -\dfrac{\pi}{6},\dfrac{\pi}{6}\right]
,\left[  \dfrac{-(k+15)\pi}{8},\dfrac{-(k+9)\pi}{8}\right]  \text{ },\left[
\dfrac{(k+9)\pi}{8},\dfrac{(k+15)\pi}{8}\right]  \text{con }%
k=0,6,12,18,...\right\}  $

\ La funci\'{o}n $\ f(x)=\cos(2x)$ es estrictamente positiva en \newline\qquad
a) $\left\{  \left(  -\dfrac{\pi}{4},\dfrac{\pi}{4}\right)  ,\left(
\dfrac{-(k+10)\pi}{8},\dfrac{-(k+6)\pi}{8}\right)  \text{ },\left(
\dfrac{(k+6)\pi}{8},\dfrac{(k+10)\pi}{8}\right)  \text{con }%
k=0,8,16,24,...\right\}  $\newline\qquad b) $\left\{  \left(  -\dfrac{\pi}%
{4},\dfrac{\pi}{4}\right)  ,\left(  \dfrac{-(k+10)\pi}{2},\dfrac{-(k+6)\pi}%
{2}\right)  \text{ },\left(  \dfrac{(k+6)\pi}{2},\dfrac{(k+10)\pi}{2}\right)
\text{con }k=0,2,4,6,...\right\}  $\newline\qquad c) $\left\{  \left[
-\dfrac{\pi}{4},\dfrac{\pi}{4}\right]  ,\left[  \dfrac{-(k+10)\pi}{4}%
,\dfrac{-(k+6)\pi}{4}\right]  \text{ },\left[  \dfrac{(k+6)\pi}{4}%
,\dfrac{(k+10)\pi}{4}\right]  \text{con }k=0,1,2,3,...\right\}  $%
\newline\qquad d) $\left\{  \left(  -\dfrac{\pi}{4},\dfrac{\pi}{4}\right)
,\left(  \dfrac{-(k+10)\pi}{4},\dfrac{-(k+6)\pi}{4}\right)  \text{ },\left(
\dfrac{(k+6)\pi}{4},\dfrac{(k+10)\pi}{4}\right)  \text{con }%
k=0,16,32,48,...\right\}  $

\ La funci\'{o}n $\ f(x)=\cos\left(  \dfrac{1}{4}x\right)  $ es estrictamente
positiva en \newline\qquad\medskip a) $\left\{  (-2\pi,2\pi),\left(
-(k+\dfrac{5}{2})8\pi,-(k+\dfrac{3}{2})8\pi\right)  ,\left(  (k+\dfrac{3}%
{2})8\pi,(k+\dfrac{5}{2})8\pi\right)  \text{ con }k=0,1,2,3,...\right\}
$\newline\qquad b) $\left\{  \left[  -2\pi,2\pi\right]  ,\left[  -(k+\dfrac
{5}{2})8\pi,-(k+\dfrac{3}{2})8\pi\right]  ,\left[  (k+\dfrac{3}{2}%
)8\pi,(k+\dfrac{5}{2})8\pi\right]  \text{ con }k=0,2,4,6,...\right\}
$\newline\qquad c) $\left\{  (-2\pi,2\pi),\left(  -(k+\dfrac{5}{4}%
)8\pi,-(k+\dfrac{3}{4})8\pi\right)  ,\left(  (k+\dfrac{3}{4})8\pi,(k+\dfrac
{5}{4})8\pi\right)  \text{ con }k=0,4,8,12,...\right\}  $\newline\qquad d)
$\left\{  (-2\pi,2\pi),\left(  -(k+\dfrac{5}{4})8\pi,-(k+\dfrac{3}{4}%
)8\pi\right)  ,\left(  (k+\dfrac{3}{4})8\pi,(k+\dfrac{5}{4})8\pi\right)
\text{ con }k=0,3,6,9,...\right\}  $

\ La funci\'{o}n $\ f(x)=\cos(x)$ es estrictamente positiva en \newline\qquad
a) $\left\{  \left(  -\dfrac{\pi}{2},\dfrac{\pi}{2}\right)  ,\left(
\dfrac{-(k+5)\pi}{2},\dfrac{-(k+3)\pi}{2}\right)  \text{ },\left(
\dfrac{(k+3)\pi}{2},\dfrac{(k+5)\pi}{2}\right)  \text{con }%
k=0,4,8,12,...\right\}  $\newline\qquad b)$\left\{  \left(  -\dfrac{\pi}%
{2},\dfrac{\pi}{2}\right)  ,\left(  \dfrac{-(k+5)\pi}{4},\dfrac{-(k+3)\pi}%
{4}\right)  \text{ },\left(  \dfrac{(k+3)\pi}{4},\dfrac{(k+5)\pi}{4}\right)
\text{con }k=0,3,6,9,...\right\}  $ \newline\qquad c) $\left\{  \left(
-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)  ,\left(  -(k+5)\pi,-(k+3)\pi\right)
\text{ },\left(  (k+3)\pi,(k+5)\pi\right)  \text{con }k=0,1,2,3,...\right\}
$\newline\qquad d) $\left\{  \left[  -\dfrac{\pi}{2},\dfrac{\pi}{2}\right]
,\left[  \dfrac{-(k+5)\pi}{2},\dfrac{-(k+3)\pi}{2}\right]  \text{ },\left[
\dfrac{(k+3)\pi}{2},\dfrac{(k+5)\pi}{2}\right]  \text{con }%
k=0,2,4,6,...\right\}  $

\ La funci\'{o}n $\ f(x)=\sin(3x)$ es estrictamente positiva en \newline\qquad
a) $\left\{  \left(  \dfrac{-(k+6)\pi}{9},\dfrac{-(k+3)\pi}{9}\right)  \text{
},\left(  \dfrac{k\pi}{9},\dfrac{(k+3)\pi}{9}\right)  \text{con }%
k=0,6,12,18,...\right\}  $\newline\qquad b) $\left\{  \left(  \dfrac
{-(k+6)\pi}{6},\dfrac{-(k+3)\pi}{6}\right)  \text{ },\left(  \dfrac{k\pi}%
{6},\dfrac{(k+3)\pi}{6}\right)  \text{con }k=0,3,6,9,...\right\}  $%
\newline\qquad c) $\left\{  \left[  \dfrac{-(k+6)\pi}{3},\dfrac{-(k+3)\pi}%
{3}\right]  \text{ },\left[  \dfrac{k\pi}{3},\dfrac{(k+3)\pi}{3}\right]
\text{con }k=0,2,4,6,...\right\}  $\newline\qquad d) $\left\{  \left(
\dfrac{-(k+6)\pi}{6},\dfrac{-(k+3)\pi}{6}\right)  \text{ },\left(  \dfrac
{k\pi}{6},\dfrac{(k+3)\pi}{6}\right)  \text{con }k=0,1,2,3,...\right\}  $

\ La funci\'{o}n $\ f(x)=\sin(x)$ es estrictamente positiva en \newline%
\qquad\medskip a) $\left\{  \left(  -(k+2)\pi,-(k+1)\pi\right)  \text{
},\left(  k\pi,(k+1)\pi\right)  \text{con }k=0,2,4,6,...\right\}  $%
\newline\qquad b) $\left\{  \left(  -k\pi,-(k-1)\pi\right)  \text{ },\left(
k\pi,(k+1)\pi\right)  \text{con }k=0,4,8,12,...\right\}  $\newline\qquad c)
$\left\{  \left(  -(k-2)\pi,-(k+3)\pi\right)  \text{ },\left(  k\pi
,(k+1)\pi\right)  \text{con }k=0,1,2,3,...\right\}  $\newline\qquad d)
$\left\{  \left[  -(k+2)\pi,-(k+1)\pi\right]  \text{ },\left[  k\pi
,(k+1)\pi\right]  \text{con }k=0,3,6,9,...\right\}  $

\ La funci\'{o}n $\ f(x)=\sin(4x)$ es estrictamente positiva en \newline\qquad
a)$\left\{  \left(  \dfrac{-(k+8)\pi}{16},\dfrac{-(k+4)\pi}{16}\right)  \text{
},\left(  \dfrac{k\pi}{16},\dfrac{(k+4)\pi}{16}\right)  \text{con
}k=0,8,16,24,...\right\}  $\newline\qquad b) $\left\{  \left[  \dfrac
{-(k+8)\pi}{2},\dfrac{-(k+4)\pi}{2}\right]  \text{ },\left[  \dfrac{k\pi}%
{2},\dfrac{(k+4)\pi}{2}\right]  \text{con }k=0,1,2,3,...\right\}  $
\newline\qquad c) $\left\{  \left(  \dfrac{-(k+8)\pi}{8},\dfrac{-(k+4)\pi}%
{8}\right)  \text{ },\left(  \dfrac{k\pi}{8},\dfrac{(k+4)\pi}{8}\right)
\text{con }k=0,4,8,12,...\right\}  $\newline\qquad d) $\left\{  \left(
\dfrac{-(k+8)\pi}{4},\dfrac{-(k+4)\pi}{4}\right)  \text{ },\left(  \dfrac
{k\pi}{4},\dfrac{(k+4)\pi}{4}\right)  \text{con }k=0,2,4,6,...\right\}  $

\ La funci\'{o}n $\ f(x)=\sin(2x)$ es estrictamente positiva en \newline\qquad
a) $\left\{  \left(  \dfrac{-(k+4)\pi}{4},\dfrac{-(k+2)\pi}{4}\right)  \text{
},\left(  \dfrac{k\pi}{4},\dfrac{(k+2)\pi}{4}\right)  \text{con }%
k=0,4,8,12,...\right\}  $\newline\qquad b) $\left\{  \left[  \dfrac{-k\pi}%
{4},\dfrac{-(k-1)\pi}{4}\right]  \text{ },\left[  \dfrac{k\pi}{4}%
,\dfrac{(k+1)\pi}{4}\right]  \text{con }k=0,2,4,6,...\right\}  $\newline\qquad
c)$\left\{  \left(  \dfrac{-k\pi}{4},\dfrac{-(k-1)\pi}{4}\right)  \text{
},\left(  \dfrac{k\pi}{4},\dfrac{(k+1)\pi}{4}\right)  \text{con }%
k=0,1,2,3,...\right\}  $ \newline\qquad d) $\left\{  \left(  \dfrac{-k\pi}%
{2},\dfrac{-(k-1)\pi}{2}\right)  \text{ },\left(  \dfrac{k\pi}{2}%
,\dfrac{(k+1)\pi}{2}\right)  \text{con }k=0,2,4,6,...\right\}  $

\ La funci\'{o}n $\ f(x)=\sin\left(  \dfrac{1}{2}x\right)  $ es estrictamente
positiva en \newline\qquad a)$\left\{  \left(  -(k+1)4\pi,-(k+\dfrac{1}%
{2})4\pi\right)  \text{ },\left(  k4\pi,(k+\dfrac{1}{2})4\pi\right)  \text{con
}k=0,1,2,3,...\right\}  $\newline\qquad b) $\left\{  \left[  -(k+1)4\pi
,-(k+1)4\pi\right]  \text{ },\left[  k4\pi,(k+1)4\pi\right]  \text{con
}k=0,3,6,9,...\right\}  $\newline\qquad c) $\left\{  \left(  -(k+1)2\pi
,-(k+1)2\pi\right)  \text{ },\left(  k2\pi,(k+1)2\pi\right)  \text{con
}k=0,2,4,6,...\right\}  $\newline\qquad d) $\left\{  \left(  -(k+1)2\pi
,-(k+\dfrac{1}{4})2\pi\right)  \text{ },\left(  k2\pi,(k+\dfrac{1}{4}%
)2\pi\right)  \text{con }k=0,4,8,12,...\right\}  $


\end{document}